Check your baseboards and electrical outlets for leaks (remove the covers for this).

I agree with the PP that it sounds like somethig else.

It's not the fireplace if it's all insulated and taped up. Sounds like bad attic and or wall insulation.

Get an oscillating ceramic heater for the room.

What can I do about my drafty fireplace. The family room is the coldest room in the house. I have one of those balloon plugs in the flue (does not help) the fireplace is taped up with insulating tape, and there is a blanket hanging on top of it. Still the room is very cold. There are two vents in the room and the air is coming out pretty warm. What else can I do about this fireplace?